What might a second summit achieve?

Analysis by Laura Bicker, BBC News, Seoul

Mr Trump's goal will be to extract pledges from Kim Jong-un without giving too much ground. The Trump administration has said it is not willing to lift sanctions, but it has mentioned helping out the North's economy.

However, handing over such aid to a secretive state which has yet to declare a list of its weapons facilities or allow in independent inspectors is bound to raise more than eyebrows.

So Mr Trump has to extract a written pledge from Mr Kim. Otherwise these summits will be seen as all show, and very little substance.

As for Mr Kim's bargaining chips, we have been told he could be prepared to give up his nuclear production site known as Yongbyon.

I've also been told by some sources close to Pyongyang that Mr Kim does want to achieve something his father and grandfather never did. A peace treaty.

The prospect of becoming the US president who ended the 68-year long Korean War is bound to be a tantalising one for Mr Trump.

What did he say about unity?

After two years of rancorous partisanship, Mr Trump on Tuesday night repeated calls for political unity that he has made in his last two annual speeches to Congress.

"Together, we can break decades of political stalemate," he said. "We can bridge old divisions, heal old wounds, build new coalitions."

A special prosecutor is still investigating alleged Russian meddling in the 2016 presidential election and possible collusion between the Trump campaign and Russia, which the president and Moscow deny.

As Mr Trump delivered his nationally televised speech, his chief congressional antagonist was sitting at the rostrum over his shoulder.

The Democratic leader of the House, Speaker Nancy Pelosi, tweeted afterwards: "It will take days to fact-check all the misrepresentations that the president made tonight."